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# L (complexity)

In computational complexity theory, L is the complexity class containing decision problems which can be solved by a deterministic Turing machine using a logarithmic amount of memory space. Intuitively, logarithmic space is enough space to hold a constant number of pointers into the input and a logarithmic number of boolean flags.

A generalization of L is NL, which is the class of languages decidable in logarithmic space on a nondeterministic Turing machine. We then trivially have $L \subseteq NL$. Also, a decider using O(log n) space cannot use more than 2O(log n)=nO(1) time, because this is the total number of possible configurations; thus, $L \subseteq P$, where P is the class of problems solvable in deterministic polynomial time.

Every problem in L is complete under log-space reductions; since this is useless, weaker reductions are defined which allow identification of stronger complete problems in L, but there is no generally accepted definition of L-complete.

Important open problems include whether L = P, and whether L = NL.

The related class of function problems is FL. FL is often used to define logspace reductions.

A breakthrough October 2004 paper by Omer Reingold showed that USTCON, the problem of whether there exists a path between two vertices in a given undirected graph, is in L, establishing that L = SL, since USTCON is SL-complete.

One consequence of this is a simple logical characterization of L: it contains precisely those languages expressible in first order logic with an added commutative transitive closure operator (in graph theoretical terms, this turns every connected component into a clique).

## References

03-10-2013 05:06:04